Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 19

Evaluate the function $\displaystyle \lim \limits_{x \to 0} \frac{\sin x}{x + \tan x} $ at the given
numbers $ x = \pm 1, \pm 0.5, \pm 0.2, \pm 0.1, \pm 0.05, \pm 0.01 $ and guess the value of the limit, if it exists.

Substitute the given values of $x$


$
\begin{equation}
\begin{aligned}

\begin{array}{|c|c|}
\hline\\
x & f(x) \\
\hline\\
1 & 0.329033 \\
0.5 & 0.458209 \\
0.2 & 0.493331 \\
0.1 & 0.498333 \\
0.05 & 0.499583 \\
0.01 & 0.499983 \\
-0.01 & 0.499983 \\
-0.05 & 0.499583 \\
-0.1 & 0.498333 \\
-0.2 & 0.493331 \\
-0.5 & 0.458209 \\
-1 & 0.329033\\
\hline

\end{array}


\end{aligned}
\end{equation}
$


The table shows that as $x$ approaches 0 from left and right, the limit approaches a value of $\displaystyle \frac{1}{2}$
.


$\displaystyle \lim \limits_{x \to 0} \frac{\sin x}{x + \tan x} = \frac{\sin (0.000001)}{0.000001 + \tan (0.000001)} = \frac{1}{2}$